Provably accurate adaptive sampling for collocation points in physics-informed neural networks

2025-04-01

AI Summary

To address the low sampling efficiency and unreliable accuracy in Physics-Informed Neural Networks (PINNs), this paper proposes an adaptive sampling method grounded in the Hessian matrix of the PDE residual. The method establishes, for the first time, a provable integral approximation error bound for PINN sampling strategies. By integrating the geometric structure encoded in the residual’s Hessian with a novel weighted numerical quadrature scheme, it overcomes theoretical limitations inherent in uniform or heuristic sampling approaches. Leveraging automatic differentiation and physics-informed modeling, the proposed framework significantly enhances solution accuracy and convergence speed across multiple 1D and 2D PDE benchmarks. Under identical computational cost, it reduces the PDE residual by 30–50% while providing rigorous theoretical guarantees on approximation quality.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.

Problem

• Efficiently solving PDEs with adaptive sampling
• Improving collocation points selection in PINNs
• Provably accurate Hessian-based sampling method

Innovation

• Hessian-based adaptive collocation point sampling
• New quadrature method for integral approximation
• Provably accurate PDE residual minimization